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Theorem hosval 26952
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hosval  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )

Proof of Theorem hosval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hosmval 26947 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
21fveq1d 5807 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5805 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5805 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6252 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  +h  ( T `
 x ) )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
6 eqid 2402 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
7 ovex 6262 . . . 4  |-  ( ( S `  A )  +h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5888 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  +h  ( T `  A ) ) )
92, 8sylan9eq 2463 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
1093impa 1192 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    |-> cmpt 4452   -->wf 5521   ` cfv 5525  (class class class)co 6234   ~Hchil 26130    +h cva 26131    +op chos 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-hilex 26210
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-map 7379  df-hosum 26942
This theorem is referenced by:  hoscl  26957  hoaddcomi  26984  hodsi  26987  hoaddassi  26988  hocadddiri  26991  hoaddid1i  26998  honegsubi  27008  hoadddi  27015  hoadddir  27016  lnophsi  27213  hmops  27232  adjadd  27305  nmoptrii  27306  leopadd  27344  pjsdii  27367  pjscji  27382  pjtoi  27391
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